JUSHIRA THELAKKAT (jt39634)

PROBLEM 1

Use definitions to prove (𝐴−1)𝑇= (𝐴𝑇)−1, Where A is an invertible square matrix and AT means the transpose of matrix A.

Solution:

In [9]:
from IPython.display import Image
Image(filename='Ass1Q1.png')
Out[9]:

PROBLEM 2

A bank makes four kinds of loans to its personal customers and these loans yield the following annual interest rates to the bank:  First mortgage 14%  Second mortgage 20%  Home improvement 20%  Personal overdraft 10% We are interested in the bank’s lending strategy. The information we know is as following:

  1. In total $250 million is lent out.
  2. First mortgages are 55% of all mortgages (i.e. first and second mortgage) issued.
  3. Second mortgages are 25% of all loans issued.
  4. The average interest rate on all loans is 15%. Calculate the lending strategy using matrix inversion.

Solution:

In [7]:
from IPython.display import Image
Image(filename='Ass1Q2i.png')
Out[7]:

Using R to solve the above matrix multiplication,

In [1]:
from IPython.display import Image
Image(filename='Ass1Q2.png')
Out[1]:

Thus we have 76.39 million in First Mortgage, 62.5 million in Second Mortgage, 31.95 million in Home Improvement and 79.167 million in Personal Overdraft as the bank's lending strategy

PROBLEM 3

Given the current state of the labor force the company estimate that, each year, they have 100000 minutes of assembly time, 50000 minutes of polishing time and 60000 minutes of packing time available. Add the non-negative constraints which ensure a positive number of units are manufactured for each type of variant. Write down all the constraints in the form 𝐴𝑥 ≤𝑏

Solution:

In [10]:
from IPython.display import Image
Image(filename='Ass1Q3.png')
Out[10]:

PROBLEM 4

Solution:

As suggested in the question, we make the A matrix invertible, by adding a condition/equation that fixes the sum of the ratings. For simplicity, we say the ratings add up to 0. So apart from the 10 equations, we add another constraint i.e.

r1 + r2 + r3 + r4 + r5 = 0

In [12]:
from IPython.display import Image
Image(filename='Ass1Q4.png')
Out[12]:

On solving the equation given in the question, Transpose(A) 𝐴 𝑟 = Transpose(𝐴) 𝑏

We have our new A and new B and using this in R , we get

r1 = -24.8 , r2 = 18.2 , r3 = -8 , r4 = -3.4 , r5 = 18